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Find the simplified product where [tex]x \geq 0[/tex]:

[tex]\[ \sqrt{5x}\left(\sqrt{8x^2} - 2\sqrt{x}\right) \][/tex]

A. [tex]\[ \sqrt{10x} \][/tex]

B. [tex]\[ 2x\sqrt{40x} - 2x \][/tex]

C. [tex]\[ 2x\sqrt{10x} - 2\sqrt{5x} \][/tex]

D. [tex]\[ 2x\sqrt{10x} - 2x\sqrt{5} \][/tex]


Sagot :

Certainly! Let's simplify the given expression step by step. We have:

[tex]\[ \sqrt{5x} \left( \sqrt{8x^2} - 2\sqrt{x} \right) \][/tex]

First, let’s handle the terms inside the parentheses separately:

1. Simplify [tex]\(\sqrt{8x^2}\)[/tex]:
[tex]\[ \sqrt{8x^2} = \sqrt{4 \cdot 2 \cdot x^2} = \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{x^2} = 2\sqrt{2} \cdot x \][/tex]

2. Simplify [tex]\(\sqrt{x}\)[/tex]:
This is already in its simplest form as [tex]\(\sqrt{x}\)[/tex].

So now the expression inside the parentheses becomes:
[tex]\[ 2\sqrt{2} \cdot x - 2\sqrt{x} \][/tex]

Next, we distribute [tex]\(\sqrt{5x}\)[/tex] across the terms inside the parentheses:

[tex]\[ \sqrt{5x} \cdot \left( 2\sqrt{2} \cdot x - 2\sqrt{x} \right) \][/tex]

We do this term-by-term:

1. Simplify [tex]\(\sqrt{5x} \cdot 2\sqrt{2} \cdot x\)[/tex]:
[tex]\[ \sqrt{5x} \cdot 2\sqrt{2} \cdot x = 2 \sqrt{10x^2} \cdot x = 2 \sqrt{10} \cdot |x| \cdot x \][/tex]
Since [tex]\(x \geq 0\)[/tex], [tex]\(|x| = x\)[/tex]. So, this simplifies to:
[tex]\[ 2 \cdot \sqrt{10} \cdot x \cdot x = 2\sqrt{10} \cdot x^2 \][/tex]

2. Simplify [tex]\(\sqrt{5x} \cdot -2\sqrt{x}\)[/tex]:
[tex]\[ \sqrt{5x} \cdot -2\sqrt{x} = -2 \cdot \sqrt{5x \cdot x} = -2 \cdot \sqrt{5x^2} = -2 \cdot \sqrt{5} \cdot |x| \][/tex]
Again, since [tex]\(x \geq 0\)[/tex], [tex]\(|x| = x\)[/tex]. So, this simplifies to:
[tex]\[ -2 \cdot \sqrt{5} \cdot x \][/tex]

Putting these pieces together, the entire expression simplifies to:
[tex]\[ 2\sqrt{10} \cdot x^2 - 2\sqrt{5} \cdot x \][/tex]

Therefore, the simplified product is:

[tex]\[ 2\sqrt{10} \cdot x^{3/2} - 2\sqrt{5} \cdot x = 2\sqrt{10} \cdot x \cdot x - 2\sqrt{5} x \][/tex]

So the correct choice is:

[tex]\[ 2 x \sqrt{10 x}-2 x \sqrt{5} \][/tex]

So the correct answer from the given options is:
[tex]\[ 2x\sqrt{10x} - 2x\sqrt{5} \][/tex]